a modular framework for performance-based durability
TRANSCRIPT
A Modular Framework for Performance-BasedDurability Engineering: from Exposure to Impacts
Madeleine M. Flinta,∗, Jack W. Bakera, Sarah L. Billingtona
aStanford University, Department of Civil and Environmental Engineering, Stanford, CA,USA
Abstract
A modular framework for assessing the economic, environmental, and socialimpacts of structural durability has been proposed and applied to a concretestructure expected to undergo climate-change-accelerated chloride-induced re-inforcement corrosion. The proposed performance-based durability engineering(PBDE) framework comprehensively considers uncertainty, accommodates non-stationary exposure, and computes quantitative sustainability metrics. Drawingon previous work in the nuclear risk and earthquake engineering communities,PBDE’s three analysis stages are de-coupled at pinch-points, allowing the use ofa convolution integral to link uncertainty in exposure, deterioration and repair,and sustainability impacts. The convolution-based methodology for the PBDEframework has been compared with traditional Monte Carlo simulation. Resultsof the convolution approach were statistically equivalent to brute-force MonteCarlo analysis using the same number of simulations, and the convolution ap-proach has advantages in deaggregation, backwards conditioning, and updatingof results to reflect new information or models. Limitations of the convolutionapproach are discussed, as are possible techniques for decreasing computationalexpense and areas for future work. Potential applications for PBDE includedesign code calibration, decision support for climate change adaptation policy,and sensitivity assessment to direct research.
Keywords:durability, sustainability, performance-based engineering
1. Introduction
The design and renewal of sustainable and resilient infrastructure pose chal-lenges to managers operating under funding constraints and changing use andexposure conditions. Numerous studies note a backlog of infrastructure re-newal in the United States; reducing this backlog demands efficient allocation
∗Corresponding authorEmail addresses: [email protected] (Madeleine M. Flint ), [email protected]
(Jack W. Baker), [email protected] (Sarah L. Billington)
Preprint submitted to Structural Safety May 6, 2014
of resources to address the most critical structures [1]. In the US transporta-tion sector, the cost of rehabilitating approximately 67,000 structurally deficientbridges is estimated at $33 billion USD [2]. Approximately 80% of all US bridgesand 50% of structurally deficient bridges were constructed prior to 1960, whichsuggests likely growth of the current backlog without implementation of a sub-stantial preventive maintenance program.
The need for additional infrastructure renewal to adapt to the changingclimate will further stretch resources, and will require balancing costs, envi-ronmental impacts, and societal risk. Climate change is predicted to increasethe rate of damage caused by corrosion of black steel reinforcement in concretestructures [3–6], the most common cause of reinforced concrete deterioration[7]. In the US, over 65% of all bridges and 34% of structurally deficient bridgeshave reinforced or prestressed concrete substructures and superstructures; manyother bridges have reinforced concrete decks [2]. Cement and steel production inthe US were estimated to emit 96.5 million metric tons of greenhouse gases (CO2
eq.) in 2011 [8], and cement production comprises approximately 5% of annualworldwide anthropogenic carbon dioxide emissions [9]. The high environmentalimpacts of infrastructure materials create a self-reinforcing cycle: unsustainableinfrastructure design worsens climate change, thereby reducing infrastructuredurability, resiliency, and sustainability, and spurring further, unsustainable, in-frastructure renewal. To support rational infrastructure management, federal,state, and municipal agencies in the United States have demanded quantitativeperformance metrics for sustainability [10]. Decision-makers must aim to ensurea low and acceptable degree of natural hazards risk while meeting sustainabil-ity goals, which in turn requires consideration of uncertainty in efforts to curbemissions, climate sensitivity, and the impact of climate change on structuraldurability and performance.
Due to the importance of providing decision support for infrastructure de-sign, management, and rehabilitation, many approaches to durability designhave recently been proposed or refined, e.g., [4; 6; 11–17] for reinforced con-crete structures. Existing durability design approaches balance merits and con-straints; lack of applicability to multiple types of infrastructure, inability to se-lect from a variety of models, and inadequate consideration of climate change arecommon limitations. Several approaches currently employed by transportationinfrastructure managers may not accurately predict deterioration in changinguse and exposure conditions. These methods are also in many cases insuffi-cient to assess innovative designs or repair strategies. While researchers haveaddressed some limitations, many existing research approaches do not extendto the fully-probabilistic economic, social, and environmental decision supportdata required by operators. A more general and flexible methodology is neededto tie together research models in a decision-oriented approach.
A framework is proposed herein to circumvent common limitations andto provide a comprehensive methodology for structural durability assessment.Drawing on related methodologies used in probabilistic risk assessment for earth-quake engineering, a performance-based approach is taken [18–20]. Performance-based approaches emphasize the direct computation of engineering variables of
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(1) Exposure Analysis
(2a) Deterio-ration
Analysis
(2b) Repair
Analysis
(3) Impact
Analysis
Exposure Conditions
(EC)
e.g., local air temperature
Damage Measures
(DM)
e.g., delamination
Timing of Repair Actions
(tRA)
e.g., cathodic protection
Decision Information
(DI)
e.g., cost, safety,
downtime, energy use
Figure 1: Stages and pinch-points of the PBDE assessment. Simulations are performed instages based on a limited number of pinch-point variables passed on from the preceding anal-ysis stage.
interest through the use of mathematical models, rather than the use of pre-scriptive approaches intended to ensure acceptable performance. The proposedframework considers uncertainty in all analysis stages, similar to many riskframeworks in other application domains, yet distinctly new relative to the ma-jority of durability frameworks.
As first proposed in [21], the framework offers conceptual clarity in assessinginfrastructure performance by separating the contributions of multiple disci-plines into discrete analysis stages. Robust decision support is provided bycombining the uncertainty associated with each analysis stage. The frameworklinks a series of conditionally independent analysis stages at “pinch-points,”where only a few variables are passed from one stage to the next, as shown inFig. 1. Stage (1), exposure analysis, links assumptions about future emissionsand weather patterns to simulations of exposure conditions, such as air tem-perature and precipitation, at the site. Deterioration analysis, stage (2a), usesexposure condition time series to predict the evolution of detectable damagemeasures, such as the presence of cracking, over time. These damage measuresare passed to repair analysis, stage (2b), for simulated inspections to determinewhether or not a repair action, such as cathodic protection or surface treat-ment, is applied. Finally, in stage (3), impact analysis, an inventory of thecosts, downtime, and environmental impacts of the different repair actions isused to calculate decision data for combinations of repair timing.
The proposed framework can be applied to different structures and expo-sures, and its modularity allows models to be interchanged within each analysisstage with minimal effect on other stages. The comprehensive inclusion of un-certainty from exposure to final decision data can lead to the accumulation oflarge uncertainties in output decision information. However, this uncertaintycan be traced back to its origin through deaggregation and backwards condi-tioning, and the approach accurately accounts for the randomness inherent toinfrastructure performance. PBDE may be preferred to other approaches whenit is desirable to trace the influence of several sources of uncertainty, to com-pare the predictions of multiple climate or deterioration models, or to developfully-probabilistic descriptions of multi-attribute decision data.
This paper discusses the background of the proposed PBDE framework andother durability engineering approaches. Descriptions of the methodology andeach analysis stage are followed by an example to illustrate application of the
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framework. Results obtained using the proposed convolution-based methodol-ogy for the PBDE framework are compared to results using a traditional MonteCarlo approach. Finally, conclusions regarding the usefulness of the methodol-ogy, anticipated limitations, and future work are discussed.
2. Background
The proposed framework relates to a broad context of research, includingreliability and risk theory, and decision-support methods. Previous applica-tions of performance-based design to durability have focused on service life asa proxy for performance, e.g., [14; 17], or the optimization of more direct per-formance metrics such as costs [15]. The proposed framework is analogous toother methodologies developed for performance-based engineering and design,especially those targeting nuclear risk and earthquake engineering.
2.1. Durability engineering methodologies
Durability engineering methodologies share the goal of improving decisionsmade in the design of new structures and repair of existing structures, butdiffer widely along three measures. The approaches are split between thosethat compute direct decision-making information such as cost, and others thatconsider implicit metrics such as service life. Approaches from both implicitand explicit decision orientations use physics-based, empirical, or survey-deriveddeterioration models. Some approaches incorporate uncertainty whereas othersare deterministic. Background on various approaches is given in the followingsections according to the major distinction between service life prediction andoptimization of decision-making information.
2.1.1. Service life prediction
Service life prediction methodologies attempt to ensure acceptable structuraldurability by calculating expected service life for different geometries, materials,and inspection and maintenance strategies. Methods of modeling deteriorationand service life range from empirical models derived from surveyed conditiondata to coupled, physics-based, numerical models. Practitioner-oriented guidesto design and rehabilitation tend to use analytical models, but vary from lim-ited inclusion of uncertainty, e.g., [16; 22], to full consideration, e.g., [17]. Loadand resistance factor design (LRFD) [11], a variety of reliability methods, in-cluding first order reliability method (FORM) [23], Monte Carlo assessment [6],and fuzzy set theory [14], among many others, have recently been proposed asdurability design frameworks. These approaches may focus on adequate mate-rial design [14], or at a combination of material, geometry, and efforts to limitexposure [24]. Some approaches consider climate change scenarios, e.g., [3–6],or structural safety, e.g., [6; 23], which are also included within the scope of theproposed PBDE framework.
Approaches within service life prediction vary in strengths and limitations.Using service life as a proxy for performance is stable: decision results are not
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dependent on cost fluctuations or discount rates. Approaches pairing simple de-terioration modeling with deterministic assessment are easy to implement andprovide preliminary design guidance. Advanced models are capable of simu-lating service life under a wide variety of exposures, and can be used to assessinnovative materials and repair actions. However, simple models may be inaccu-rate compared to more sophisticated models, whereas advanced models requirea greater number of parameters that may not be readily available.
2.1.2. Maintenance and repair optimization
Maintenance and repair optimization approaches focus on decision informa-tion such as costs, environmental impacts, and safety, or utility functions of thisinformation. Many recent approaches propose methods for optimizing designand maintenance of reinforced concrete infrastructure, including through useof Markov chains [12], Bayesian nets [13], renewal theory [25] and cellular au-tomata [26], among others. These approaches frequently utilize probabilistic de-scriptions of deterioration derived from surveys of existing structures, and mayrequire large datasets to produce robust predictions. Discrete Markov chains,which link a damage state at one time step to the probability of moving into amore severe damage state at the next time step, are commonly used in bridgemanagement system software [27]. A hybrid approach has been proposed in [28]that combines the physics-based modeling of deterioration commonly used inservice life prediction with a Markovian approach more suited to maintenanceoptimization.
Like service life prediction approaches, maintenance and repair optimizationmethods balance strengths and limitations. In general, these optimization ap-proaches are practical: they focus on the development of decision informationneeded to select maintenance and repair strategies, and can be used to modela variety of structural elements. However, deterioration models derived fromsurvey data may not be robust when data is limited, or when new climates ormaterials are studied. While hybrid approaches, e.g., [28], are in theory capableof optimizing maintenance in a changing climate, few studies consider both cli-mate and impact uncertainty as proposed in PBDE. Furthermore, in some casesthe focus on decision optimization leads to scarce availability of intermediateinformation for validation of deterioration predictions. Given the difficulty ofmanaging large infrastructure networks, maintenance and repair optimizationmethods offer valuable insight into possibilities for improving network-level per-formance.
2.2. Performance-based risk engineering methodologies
Performance-based methods directly evaluate life-cycle outcomes by pre-dicting and quantifying the range of behavior of engineered systems. Manyprobabilistic risk assessment methodologies compute decision-making informa-tion by combining conditionally independent analyses. Nuclear risk engineeringmethodologies, e.g., [18], generally separate an assessment into several analysisstages, incorporate uncertainty in each stage, and propagate uncertainty to ob-tain descriptions of decision variables. These techniques were further developed
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in performance-based earthquake engineering (PBEE), where they have beenused to update building codes, design special structures, and to conduct largescale studies for use in policy [19; 20]. Performance-based methodologies forearthquake developed by the Pacific Earthquake Engineering Research Center(PEER) [19] have been adapted to wind [29] and other natural hazards, and areclosely related to the PBDE framework proposed herein. These performance-based frameworks are significant for their usefulness in guiding code design andpolicy, and for their identification of future research needs. In applying PEERperformance-based approaches to climate-influenced natural hazards, care mustbe taken to assure that the assumptions used to develop a hazard curve are con-sistent with non-stationary extreme event risk; the proposed PBDE frameworksupports consideration of non-stationary hazards. Other recent work has linkedperformance-based assessment of sustainability and natural hazard risk, includ-ing assessment of repair practices [30]. While performance-based probabilisticrisk assessment is but one of many methods for managing risk [31], it has beenwidely and successfully used in civil engineering.
3. Proposed probabilistic framework
The proposed methodology for performance-based durability engineering an-alyzes a given structure and site in three stages: (1) exposure, (2a) deteriorationand (2b) repair, and (3) impact. The results of these stages are sets of gener-alized “pinch-point” variables: exposure conditions (EC), damage measures(DM), repair action timing (tRA), and decision information (DI), respectively.Deterioration and repair analysis occur in the same stage due to the dependencyof deterioration on previous repair actions. All analysis stages may give rise tolarge uncertainties, both aleatory and epistemic, which are incorporated intothe full probabilistic assessment.
Each analysis stage computes a complementary cumulative distribution, de-noted G, of the probability of exceeding a value of the considered pinch-pointvariable. For deterioration, repair, and impact analyses, these distributions areconditioned on a given value of the pinch-point variable output from the prioranalysis stage. Convolving a series of these conditional distributions using Eq.1 yields a final distribution for lifetime decision information.
GDI(di) =
∫∫GDI|tRA
(di|tra) |dGtRA|EC(tra|ec)| |dGEC(ec)| (1)
The incorporation of initial construction decision information, e.g., in orderto compare alternative new designs that vary in up-front and long-term expectedcosts, is possible through an additional convolution integral or through MonteCarlo simulation. Use of the convolution integral assumes that initial decisioninformation distributions are independent of repair-related decision information,as in Eq. 2, where f denotes a probability density function (PDF). If the distri-butions are not independent, Monte Carlo sampling of the initial constructionand repair-related decision information distributions allows computation of thefull lifetime distribution.
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GDI(di) =
∫GDI,t>0(di− z)fDI,t=0(z) dz (2)
The following descriptions of the analysis setup and stages refer primarily tomodels required for predicting chloride-induced corrosion in coastal reinforcedconcrete bridges, though the methodology is not limited to this case. Providedthat appropriate deterioration models are available, other structures and mecha-nisms may be studied, including wood rot in residential housing, facade damageof historic buildings, corrosion of steel infrastructure, or any other mechanismin which damage accumulates over time.
3.1. Analysis setup
PBDE assessments require many parameters and assumptions, including:structural geometry, material properties, weather data, deterioration mecha-nisms likely to affect the structure, and inspection and repair practices. Keydecisions in analysis setup include the selection of appropriate deteriorationmechanisms and related models, and the determination of the scope of uncer-tainty to be included. Treating model parameters, material properties, andstructural geometry as random variables may greatly increase the number ofsimulations required in deterioration and repair analysis. Preliminary studiesconducted during analysis setup determine an appropriate discretization of thepinch-point variables and the number of simulations required in each analysisstage.
3.2. Exposure analysis
Exposure analysis simulates local weather exposure conditions (ECs) at thestudied site. In addition to meteorological data, other causes of deterioration,including applied loads, can also be modeled, e.g., to study the interaction ofload-induced cracking and transport of aggressive ions. The models used in ex-posure analysis may take both global-scale inputs, e.g., greenhouse gas emissionscenarios, and local-scale inputs, e.g., ground surface roughness near the site.The most sophisticated analyses would include emissions models, a global cli-mate model ensemble, a regional climate model, and models that translate localweather conditions to exposure conditions acting on the structural surface, e.g.,wind-driven rain models. The outputs from the exposure models include both“records” (time series) of exposure to be used in deterioration analysis, and aprobabilistic distribution of the likelihood of experiencing a record consistentwith a characteristic level of severity, i.e., a characteristic exposure distribution(GEC). Fig. 2 illustrates components of a characteristic exposure condition:
baseline mean annual temperature, described by a generic variable, λ, and an-nual temperature “noise” and trend terms. The three terms are summed tocompute the characteristic exposure condition, increase in mean annual temper-ature over the lifetime. A distribution of the characteristic exposure conditioncan be obtained by weighting different emissions scenarios, e.g., from [32], or byfitting a distribution to the output of a climate model.
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Figure 2: Representation of example exposure analysis with characteristic distribution ofincrease in mean annual temperature and exposure “record” components.
Selection of an efficient and sufficient characteristic condition can presentdifficulties. The characteristic exposure condition must allow for simulation ofother exposure condition records consistent with a sampled value. If, for ex-ample, the selected characteristic condition is increase in mean annual globaltemperature, records of local temperature, humidity, precipitation, etc., shouldbe simulated based on assumptions consistent with the sampled value of themean temperature increase. Due to the large variability in local weather, it islikely that a suite of exposure condition records consistent with a single valueof the characteristic EC is needed to capture exposure uncertainty. An effi-cient characteristic exposure condition minimizes damage and repair variabilityat a given level of exposure, thereby limiting the number of computationally-expensive deterioration/repair simulations. Substantial research in the PBEEcommunity has studied methods for the selection of that framework’s analogouspinch-point hazard variable, and has proposed several alternative candidates[33].
3.3. Deterioration analysis
Deterioration analysis relates exposure at the site to observable or detectablestructural damage. Records of exposure conditions are passed to one or moreempirical or physical deterioration models to simulate the accumulation of ma-terial and component degradation. Degradation is linked to damage measuresselected from common visual or special inspection methods, which act as indi-cators, or thresholds, to trigger repair actions in the concurrent repair anal-ysis stage. Deterioration analysis outputs simulations of damage measures,and conditional distributions of the probability of exceeding damage over time(GDM |EC(dm, t|ec)). These distributions are not included in the calculation ofdecision information, but are used in safety and reliability assessments, and aregenerally of interest to maintenance engineers.
Several challenges are present in deterioration modeling, including spatialcorrelation and multi-mechanism issues discussed in Sec. 3.6, and other issues
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surrounding model selection and the simulation of boundary conditions. In real-ity, deterioration mechanisms may be tightly coupled, and few physical modelscapture the interdependent nature of structural deterioration. Furthermore,physics-based transport models require methods to compute nonlinear heat,moisture, and ion surface boundary conditions. Considerable research withinthe building physics community has focused on the development of methods tocompute heat and moisture surface conditions [34], which have not been widelyimplemented in infrastructure durability research [6]. Sensitivity of infrastruc-ture deterioration to boundary conditions depends on the sophistication of heatand mass transport and damage models used, which range in complexity anddegree of coupling. In the modeling of corrosion damage in reinforced concrete,this range extends from use of apparent Fickian diffusion for chloride transport[24], to numerical models that fully couple heat and mass transport with cor-rosion, e.g., [35]. An intermediate de-coupled numerical approach was used todevelop the illustration presented in Sec. 4. In addition to the differing require-ments for exposure data resolution, model selection typically involves tradeoffsbetween accuracy, computational expense, ease-of-use, and availability of re-quired material parameters. Discrepancies in predictions of deterioration ratesand sensitivity to boundary conditions between different types of heat and masstransport models have been found in [36]; the proposed PBDE framework offersa means to assess the importance of these discrepancies through comparison ofdecision outcomes.
3.4. Repair analysis
Repair analysis captures the decision-making process of operators in orderto determine when and how a damaged structure or element is repaired. In ad-dition to the damage measure simulations, inputs to repair analysis include theproposed inspection timing and practices, a set of possible repair actions, andinformation on conditions that will lead to the selection of a particular repairaction. In the method of repair analysis currently used, the input informationis mapped onto a decision tree relating damage measures observed during aninspection event to the selection of a repair action. The decision tree used in theillustration example, that of a coastal reinforced concrete structure, is shown inFig. 3. While records of repair events are available from repair analysis, theprimary output from this stage is the conditional repair action timing distribu-tion (GtRA|EC(tra|ec)) calculated from the discrete probabilities of observing aparticular combination of individual repairs, given a value of the characteristicexposure condition. For example, given two possible repair actions, a repairtiming combination, K, might be denoted tRAK
= tra1 = 10 years; tra2 = 30years.
Several factors may complicate the development of a rational and computa-tionally efficient decision tree, including the use of life-cycle costing or structuralanalysis within the decision tree, operators who do not have set repair practices,and lack of data or guidelines on what values of damage measures trigger repair.When the repair actions or damage thresholds are unclear, reviewing past in-spection and repair records may be helpful. In any case, engineering judgment
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inspection time
cathodic protection (ra1)
do nothing
delam-ination
yes
yes patch repair (ra3)
no no cathodic prevention (ra2) no
first repair
≥10y SL req’d
yes active corrosion
yes ≥20y SL req’d
no no
regular (2y interval)
special (10y interval; 32y post-CP)
yes
yes
Figure 3: Decision tree for repair actions (RA) cathodic protection (CP), cathodic preventionafter end of CP system functionality, and patch repair. Special and regular inspections occurat ten and two-year intervals, respectively. Decisions to repair are based on the damagemeasures (DM) presence of delamination and finding of an active corrosion state, as well asthe current repair state and required remaining service life (SL).
is required to develop even a “best practices” tree, which leads to a possible dis-crepancy between PBDE assessment results and actual outcomes. However, if arational management practice is planned, PBDE assessment allows comparisonof alternative strategies or designs.
3.5. Impact analysis
Impact analysis links a combination of repair action timing to the economic,environmental, and social impacts of the repairs. Economic decision informationincludes both direct measures, such as the cost of repairs, and indirect measures,such as economic loss due to building closure or traffic delays. Direct costs canbe obtained through process-based costing [37], where a cost is obtained for eachmaterial or construction action, or through cost estimates for the entire repairaction, i.e., from bid data. Environmental impacts include emission of green-house gases and other air pollutants, energy use, release of toxic chemicals, andseveral additional midpoint indicators. In many cases an inventory of materials,equipment, and energy used in repair is required for cost and environmentalimpact modeling. Indirect costs and social impacts related to structural safetyrequire additional data on use patterns. The output from impact analysis is aset of conditional distributions for decision information given repair action tim-ing (GDI|tRA
(di|tra)). If many repair actions timing combinations and decisioninformation variables are considered, the number of conditional distributionsmay become large, in which case only those conditional distributions reflectingactive combinations of repairs might be computed.
Significant sources of uncertainty are present in the calculation of repair im-pacts, including: uncertainty in which construction methods contractors will se-lect; uncertainty in quantities of materials used; variation in material resource al-location and processing worldwide; uncertainty in the future economy, includingdiscounting and inflation; uncertainty in the future energy mix and other tech-nological developments; and uncertain future populations, structure use, and oc-cupancy. For environmental impact assessment, selection of a method involvestradeoffs between sources of uncertainty: environmental input-output (EIO)
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models capture the coupled and nested intricacies of environmental impacts butcannot distinguish between local-level variations, whereas process-based life cy-cle assessment can capture local practices but requires inventory data that maynot be readily available. Hybrid approaches combine the comprehensive aspectsof economic input-output modeling with the specificity of process-based ap-proaches, and are commonly used in practice. Cass and Mukherjee [38] discussthe benefits and drawbacks of the various approaches as applied to pavements.Several recent bridge life cycle assessments have been reviewed in [39], whichcompares six environmental indicators for three real bridges of varying materialand structural form.
In addition to their uncertainty, the some impacts are likely to be correlated,e.g., an individual repair action might be more difficult to perform than average,resulting in both high costs and high closure time. PBDE assessment can deter-mine the effect of partial correlation. Given the numerous and potentially largesources of uncertainty, the analyst may prefer to take a first-order approach byassigning independent normal distributions for all initial and ongoing impacts,allowing direct calculation of (normal) impact distributions for sets of repairaction timing. This first-order method was used in the illustration presented inSection 4. With some modification, this approach could be used to assess thesensitivity of results to assumptions of future costs or technology.
3.6. Challenges and mitigation approaches
The breadth of the proposed framework leads to large computational anddata intensity, which must be mitigated to effectively perform PBDE assess-ment. Models used within exposure and deterioration analysis stages may beindividually computationally intensive, and their combination in PBDE maylead to lengthy analysis times for a single simulation. Adding to the complexityis the need to predict exposure and deterioration in multiple, disparate partsof the structure: repair analysis considers repair actions that may be selectedbased on the state of the entire structure, rather than the state of one portion ofone component. This need to consider system states requires additional effort toselect appropriate deterioration mechanisms for each component, again increas-ing computational expense. To make the PBDE assessment more tractable, theanalyst might use spatial correlation models to simulate random fields [40–42],sampling techniques [43], or other methods drawn from related performance-based methodologies.
Furthermore, in-situ exposure rarely matches the assumptions used to de-velop common single-mechanism deterioration models. Interaction of multipledeterioration mechanisms frequently accelerates overall deterioration; perform-ing separate PBDE assessments for different types of damage and adding theresults is not possible. When it is important to capture multiple deteriorationmechanisms, general deterioration approaches, e.g., the use of Markov transi-tion matrices, may be implemented within the PBDE assessment. The Marko-vian approach will yield the most realistic results when the studied structureclosely matches conditions used to derive the transition matrices, i.e., for com-mon structure materials, types, and load patterns, and when climatic conditions
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are expected to remain stable. In no case will a PBDE assessment completelydescribe all possible scenarios and impacts, but baseline results can allow forcomparison of alternative designs or repair strategies.
4. Demonstration of methodology and comparison to a Monte Carloapproach
An idealized reinforced concrete coastal structure was selected to demon-strate the use of the proposed PBDE methodology and to compare the convolu-tion method with traditional, “brute force” Monte Carlo simulation. Simplifyingassumptions increased the clarity of the illustration and minimized the compu-tational expense associated with performing both convolution and Monte Carloassessments. Section 6.4 identifies possible extensions to more sophisticated as-sumptions or models. Results included predictions of the course of damage overtime and final cost and downtime impact distributions. An additional studyused the convolution approach to analyze contributions to decision informationuncertainty, and to update decision information distributions to reflect a changein the exposure emissions scenario.
4.1. Structure, site, and deterioration mechanism
An idealized reinforced concrete coastal structure was studied for risk ofchloride-induced reinforcement corrosion. The structure was assumed to beconstructed in 2010 with planned demolition in 2090, necessitating inclusion ofpotential climate change in exposure analysis. The illustration considers highand low emissions scenarios combined with global climate uncertainty. Thechosen site of coastal Hilo, Hawaii, and functional unit of a sheltered, verticalsurface, allowed use of ambient meteorological data as surface boundary con-ditions with minimal error compared to splash zone or tidal exposure. Themoderate temperature and humidity range of Hawaii causes a high degree ofcorrosion risk while reducing the potential for multi-mechanism deterioration.A standard ordinary Portland cement concrete mix design with water-to-cementratio of 0.5 and cover depth of 50 mm was studied over a functional unit of a 1m2 surface area. The selection of a planar element minimized localization andedge effects, allowing the use of one-dimensional transport models.
4.2. Models used in analysis stages
4.2.1. Exposure analysis models
Increase in the mean annual temperature at year 2090 over the 1970 baselineserved as the characteristic exposure condition. The temperature projectionsused in this study were based on a report on climate change in the South Pacificislands, which accounted for a set of emissions scenarios representing divergenttechnology development and adaptation, economic and population growth, andfavored fuel type [32; 44]. In addition to increased mean temperature and sealevel rise, climate change is expected to induce more frequent downpour and
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extreme high sea level events in this region. Based on “high” and “low” emis-sions scenarios and climate projections, possible warming in this region rangesfrom 1.4 to 3.4 by year 2090, relative to a 1970 baseline. Increase in meanannual temperature at 2090 was modeled as the sum of equally-weighted normalrandom variables representing “high” and “low” emissions scenarios, accordingto Eq. 3, where N denotes a normal distribution. A power law was assumedfor the functional form of the climate projections in [44]; parameters for thepower law representation of temperature increase (λα) were modeled as func-tions of the characteristic exposure condition and obtained through regression.The convolution assessment discretized the exposure condition over the range1.1 to 3.5.
EC ∼ 0.5N (1.80, 0.20) + 0.5N (3.00, 0.13)
λα(tyear) = aec(tyear)nec
aec(ec) = 5.04e-3ec2 − 3.57e-2ec+ 6.49e-2
nec(ec) = 3.59e-1ec+ 3.33e-1 (3)
The characteristic exposure condition was combined with other parametersand assumptions to develop temperature, relative humidity, and chloride concen-tration surface boundary condition records representative of atmospheric coastalexposure. All surface boundary condition records were computed using Eq. 4and parameters from Table 1, where the generic variable λ represents any ofthe three types of exposure conditions. For temperature, T , local climate pa-rameters were calculated from 1970-2012 data recorded at Hilo InternationalAirport, Station ID CCOP:511492, obtained from the NOAA NCDC database(http://www.ncdc.noaa.gov/). A term (λβ) reflecting natural year-to-yearvariation in Hilo’s climate was modeled as a set of zero-mean “noise” (ε) withstandard deviation of 0.475, and added to Hilo’s baseline average mean annualtemperature (λ). The convolution assessment used the same set of indepen-dent noise records at each value of characteristic exposure, whereas the MonteCarlo assessment used a unique, independent noise record for each individualsimulation. In order to characterize seasonal temperature variation (λs), a har-monic term was added based on regression performed on the meteorologicaldata. Seasonal variations are not expected to change significantly under risingmean annual temperature in this region [44]. The approach taken herein ofadding independent terms to reflect global and local climate behavior simplifiesthe complex interaction between climate spatial scales.
λ(t) = λ+ λα(tyear) + λβ(tyear) + λs(t)
= λ+ aec(tyear)nec + ε(tyear)
+a1 sin(ωt+ b1) + a2 sin(2ωt+ b2) (4)
Annual mean relative humidity, h, and annual mean exposure to surfacechlorides, CCl, were assumed constant both year-to-year and with increasing
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Table 1: Parameters for exposure boundary condition equations
λ λ λα λβ λs
ec ε a1 b1 a2 b2T [] 23.3 1.1–3.5 ∼ N (0,0.475) 1.46 2.64 0 0h [-] 0.735 0 - 9.49e-3 -0.813 0 0CCl [%kg/kg cem] 84.6 0 - 57.2 1.24 16.5 0.172
temperature. Relative humidity seasonal variation was based on a regression onambient meteorological data. Surface chloride concentration seasonal variationwas based on a combination of survey data and a study of seasonal variationin marine aerosol salinity, details of which can found in [36]. The annual meansurface concentration was of similar magnitude to the “high” exposure in [6]and the “atmospheric coastal” exposure in [4].
4.2.2. Deterioration analysis models
Deterioration analysis modeled chloride transport, reinforcement corrosioninitiation, corrosion propagation, and concrete cover delamination, as shown inFig. 4. Use of a temperature- and moisture-dependent Fickian diffusion par-tial differential equation was amenable to the level of sophistication of exposuremodeling. The diffusion equation was solved numerically, rendering it capableof predicting transport under non-stationary boundary conditions and prevent-ing errors associated with commonly-used analytical approximations [45]. Theempirical, stochastic corrosion propagation model is a function of temperatureand chloride concentration. With the exception of corrosion initiation and de-lamination critical values, the model parameters were deterministic. While theMonte Carlo assessment sampled stochastic parameters directly from their dis-tributions, the convolution assessment used an optimized uniform Latin hyper-cube scheme. Uniform Latin hypercube offers improved sampling efficiency overbrute-force Monte Carlo in building envelope hygrothermal modeling [43]. Theroutine to determine an optimal uniform sampling scheme provided in [43] is anadaptation of the “maximin” optimization algorithm in [46]. A more detaileddescription of the deterioration modeling and required parameters is presentedin Appendix A.
4.2.3. Repair analysis models
Inspection was assumed to occur at two-year intervals from 2012 to 2080.At “regular” inspections repair analysis simulated inspection of the surface fordelamination. Every ten years (2020, 2030, . . . ) nondestructive evaluation ofcorrosion rates was simulated. Three repair options were considered: cathodicprotection, ra1, followed by cathodic prevention, ra2, if the remaining servicelife was greater than or equal to 20 years, or by patch repair, ra3, when from 10to 20 years of service were required. Titanium anode mesh cathodic protection,including removal of the concrete cover and application of repair mortar, wasassumed functional for 30 years, with 2 years of remaining protection after theend of system life [22; 47]. If indicated, installation of cathodic prevention
14
boundary conditions
chloride transport
corrosion initiation
corrosion propagation
cracking
from exposure conditions directly
1D FEM moisture/temperature-
dependent effective diffusion
stochastic, empirical
moisture/temperature-dependent semi-
empirical
stochastic, empirical
Figure 4: Schematic of deterioration models used in the demonstration.
Table 2: Impact inventory for repair actions assuming a functional unit of a 1 m2 verticalreinforced concrete surface area
Repair Action Initial Cost Ongoing Cost Downtime[USD] [USD/yr] [months]
Cathodic protection ∼ N (250, 80) ∼ N (5, 1) ∼ N (24, 6)Cathodic prevention ∼ N (150, 40) ∼ N (5, 1) ∼ N (6, 2)Patch repair ∼ N (100, 20) - ∼ N (4, 1)
after the failure of the cathodic protection system was assumed to maintainsufficient polarization of the reinforcement to prevent active corrosion initiation.The cathodic prevention system was also assumed effective for 32 years. Patchrepairs using mortar were assumed to occur over one-quarter of the functionalunit (0.25 m2). It was assumed that no repair would occur after failure of thepatch repair or cathodic prevention system, or with less than 10 years remainingservice. The repair tree yielded 51 possible combinations of repair action timing.
4.2.4. Impact analysis models
The assessment focused on two types of readily-characterized decision infor-mation: cost and downtime for repair. Cost and downtime were modeled usingestimates from [22; 47; 48]. For the assessment herein, downtime was consid-ered disruption of normal activity rather than complete loss of use, resultingin relatively long possible downtimes. All decision information was modeledas normally distributed, with means and standard deviations given in Table2. Cost and downtime were assumed stationary and uncorrelated. This as-sumption allowed direct calculation of the parameters of normally-distributeddecision information from the parameters of the individual repair impacts.
4.3. Numerical implementation
The PBDE assessment was carried out in a series of Matlab 2012b scripts.The range and discretization of time, damage measures, and decision informa-tion was determined through convergence studies. The final scheme for convo-
15
lution resulted in discretization of 10 exposure condition values and 100 dete-rioration/repair simulations per exposure condition value. The same numberof total simulations (1000) was used for both convolution and Monte Carlo ap-proaches, although more thorough investigation might have reduced the numberof simulations in either case.
Histograms were used as conditional and final damage measure distribu-tions. Empirical complementary cumulative distributions were used for allMonte Carlo approach distributions; the decision information means and stan-dard deviations were obtained directly from the simulated values. Numericalconvolution of the conditional distributions according to Eq. 1 used a trape-zoidal integration rule. The Kolmogorov-Smirnov (K-S) test, which allows sta-tistical testing of non-parametric distribution, was used to compare the con-volution and Monte Carlo results, using the one-sample or two-sample test asappropriate. Reported p-values give the probability of observing a value of themaximum difference (supremum) between two cumulative distribution functions(CDFs) based on the hypothesis that the distributions are equivalent.
5. Results
5.1. Final results
5.1.1. Exposure analysis results
Exposure analysis results reflected the primary assumption of a combinationof normal distributions for the characteristic exposure condition, increase inmean annual temperature. Exposure analysis discretized the range of the char-acteristic exposure condition into 10 values; the resulting distribution is shownin Fig. 5, along with the empirical distribution obtained from the Monte Carlosamples, and the original distribution. By the one-sample K-S test, the MonteCarlo empirical CDF was consistent with the specified distribution, which con-firmed that a sufficient number of simulations were performed for this variable(p=0.98).
5.1.2. Deterioration analysis results
Temporal evolution of the probability density function for chloride concen-tration at 50 mm is shown in Fig. 6. The contour plot was derived from theconvolved data, which branched into eight behavior modes to the Monte Carloassessment’s seven modes. From 2010-2020, the dark contour band traces thepath of all simulations as the chloride front reached 50 mm and the concentra-tion gradually increased. The wavy edges of the contours stem partially fromthe harmonic surface chloride concentration variation and partially from thedamage measure discretization. At 2020, a small number of simulations wererepaired with cathodic protection. The 2030 PDF’s tightly banded distribu-tion at 0.29 %wt/wt cement reflects variation in the mean annual temperatureprojection, including the noise term, as the apparent diffusion coefficient was afunction of deterministic parameters and therefore incapable of contributing to
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Figure 5: Characteristic exceedence distributions for increase in mean annual temperaturefrom 1970 to 2090. Distributions are presented for the “original” distribution and both con-volution discretization and Monte Carlo sampling, and for an “updated” higher-warmingscenario with convolution discretization only.
the variance. Following the trend of highest density, a set of simulations con-tinued on the same trajectory until end of service. These simulation’s relativelyhigh critical chloride concentrations prevented corrosion initiation. The 2030PDF’s large mass at zero chlorides revealed cathodic protection (CP) repair ofsome simulations at this time. Given that a CP system installed in 2030 wouldbe expected to fail in 2060, with 20 years service remaining, it was inferredthat these simulations were later repaired with cathodic protection (CPre). Asshown in the vertical PDFs at 2050 and 2070, other simulation modes includedlater cathodic protection, up until the contours with repair at 2070 and 2080,which reflect either first application of patch repair, or patch repair after fail-ure of the CP system. The trend of repair at 10-year intervals suggested thatspecial inspections for corrosion triggered repair, rather than visual delamina-tion inspection, which occurred at 2-year intervals. The small bandwidths of allcontours indicated that the separation of the behavior into the different repairmodes was due more to the stochastic corrosion initiation than variation in thecharacteristic exposure. The importance of the exposure scenario is discussedfurther in Sec. 5.3.
Corrosion rate distributions centered around relatively low rates (< 0.1µA/cm2).The corrosion rates were on the lower end of the range for airborne chloride expo-sure in [4]. Due to the low corrosion rates and early repair times, reinforcementradius loss was negligible and delamination never occurred before 2080, whichexplained the lack of repairs following regular inspections.
5.1.3. Repair analysis results
Fig. 7 shows the probability mass function (PMF), p, for the convolutionapproach’s eight and the Monte Carlo approach’s seven active combinations. In
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Figure 6: Evolution of probability density function for damage measure (DM) chloride con-centration at 50 mm as a contour plot. Contours are plotted at fDM = 1.5, 25, 50, 75, 100.Probability density functions are plotted horizontally at 1/20 scale against vertical axes atyears 2030, 2050, and 2070.
both assessments the most common combination was no repair (combination8). Several combinations consisting of only cathodic protection (combinations4-6) and three multiple-repair combinations (1-3) also occurred. The MonteCarlo simulation resulted in fewer early repairs (combinations 1-2), and higherprobability mass for combination 3. Investigation of the Latin hypercube andMonte Carlo samples indicated that several Latin hypercube critical chlorideconcentration samples were lower than the smallest Monte Carlo sample. Theseearly-initiating simulations were responsible for the convolution approach’s highprobability masses at early repairs. Despite these discrepancies, the two-sampleK-S test indicated a reasonable likelihood of distribution equivalency (p = 0.39).
5.1.4. Impact analysis results
Decision information results, shown in Fig. 8, were comprised of contribu-tions from the eight active repair combinations. The central portion of the dis-tributions differ between the two approaches, especially for cost (p=0.56,0.76for DI1, DI2). The better agreement for downtime, which had only initialimpact, suggested that the differences in repair timing probabilities for combi-nations 1-3 might have contributed to the different decision results. To deter-mine whether the discrepancy was caused by the differences in repair timingcombination probabilities or insufficient sampling of the decision variables, theMonte Carlo repair timing distribution was convolved with the convolution ap-proach’s conditional decision information distributions, yielding more robust
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Figure 7: Probability mass function for combinations of repair action timing. Combinationnumber is shown at the top of the figure.
Table 3: Decision information results for original convolution, original Monte Carlo, andconvolution with updated exposure scenario.
Decision Information Original, Con-volution
Original, MonteCarlo
Updated Expo-sure, Convolu-tion
Cost [USD] µdi1 ± σdi1 350± 217 354± 250 351± 19510% exceedence 682 656 682
Downtime[months]
µdi2 ± σdi2 17.9± 9.8 18.5± 12.9 18.0± 9.6
10% exceedence 33.2 33.1 33.2
final distributions. These distribution had better agreement with the convo-lution distribution (p=0.91 for both DI), which suggested that increasing thetotal number of Monte Carlo simulations would reduce the discrepancy. The re-maining difference between the convolution and robust Monte Carlo approachesconfirmed the role of the discrepant repair timing probabilities. Distributionmoments and selected exceedence values are contained in Table 3. A statisticaltest of the distribution moments (Cohen’s d) supported the finding of distribu-tion equivalency (effect size 0.017).
5.2. Intermediate results
5.2.1. Conditional deterioration and repair analysis results
Conditional distributions for repair action timing combinations computed forthe convolution approach are shown in Figure 9. Conditional damage measureresults were similar to the fully convolved damage measure results for eachdiscretized value of the characteristic exposure condition. The ten conditionalrepair timing PMFs were similar, with a slight trend towards late or no repair
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Figure 8: Exceedence distributions for lifetime decision information variables (a) cost and (b)downtime, for original and updated emissions scenarios. Means of the distribution are plottedas a square for the convolution approach, a circle for the Monte Carlo approach, and a trianglefor updated convolution.
in the distributions with smallest increase in mean annual temperature. Thisbehavior was consistent with the apparent diffusion model’s prediction of fasterchloride ingress at higher temperatures (see Appendix A).
5.2.2. Conditional impact analysis results
Impact analysis computed conditional distributions for cost and downtimefor all possible combinations of repair action timing. Exceedence distributionsfor five of the eight active combinations are shown in Fig. 10. Besides the zero-impact “no repair” distributions, the lowest costs were incurred by combination7, patch repair in 2080. The downtime distributions for combinations includingonly cathodic protection, as represented by combinations 3 and 4, were identical,as downtime was modeled as an initial, time-of-repair, impact. As expected,costs and downtimes were higher when multiple repairs took place, as occurredin combinations 6 and 7.
5.3. Deaggregation, backwards conditioning and updating of decision informa-tion
Whether convolution or Monte Carlo simulation is used to compute decisioninformation distributions, the modularity of the PBDE framework facilitatesstudies to deaggregate uncertainty, trace uncertainty through backwards con-ditioning, and update results to reflect new data or models. Deaggregation iscommonly used in probabilistic seismic hazard analysis to identify the contri-butions of various faults and events to the site seismic hazard. Deaggregationin PBDE can be used to assess how uncertainty in each of the three analysisstages contributes to the uncertainty in the final decision information results.Figure 11 presents deaggregation results in two ways. On the left, costs accruedover time are shown as 5%-95% probability envelopes for several combinationsof deaggregated uncertainty. Starting from a deterministic result that assumes
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Figure 9: Conditional probability mass function for combinations of repair action timing,given a value of the exposure condition (ec) mean annual temperature rise at year 2090.
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Figure 10: Conditional exceedence distributions for decision information variables (a) cost,and (b) downtime, for selected combinations (denoted in the legend with a superscript) ofrepair action timing.
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Co
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Figure 11: (a) Envelope for costs over time differentiated by types of uncertainty included.The dot-dashed line was obtained using expected values for exposure (results were identical forhigh and low warming scenarios), deterioration and repair parameters, and repair costs. Theshaded area demarcated by the solid line includes uncertainty in damage and repair param-eters (DM/tRA). The hatched area demarcated by the long-dashed line includes additionaluncertainty in repair costs (DI), for the expected exposure in the high warming scenario.The dotted area demarcated by the dotted line additionally considers the full distributionof exposure (i.e., the full PBDE assessment). (b) Complementary cumulative distributionfunctions for lifetime cost. Colors, markers and line styles are consistent with the legend in(a). Expected values for lifetime costs are plotted with a filled circle, square, and circle forno, DM/tRA, and full uncertainty, respectively. Distributions obtained using the expectedvalue for low as well as high warming are included. (c) Timeline for repair actions, includingcombination numbers. Upward-facing triangles denote cathodic protection; downward-facingtriangles denote cathodic prevention; the diamond represents patch repair.
expected values for all parameters, the relative influence of analysis stage uncer-tainty can be determined by re-computing decision results while progressivelyincluding the uncertainty in each analysis stage. In the illustration, a largeamount of cost uncertainty resulted from the inclusion of uncertainty in dam-age and repair analysis, and relatively less when uncertainty in impact analysiswas additionally considered. Virtually no additional uncertainty resulted fromincluding uncertainty in exposure analysis. On the right, the complementarycumulative distribution function also characterizes the relative importance ofuncertainty in the three analysis stages. This plot identifies how the dispersionand expected value of cost changed as additional types of uncertainty were in-cluded. The PBDE framework’s support for deaggregation through its modularstructure is a major benefit of the approach.
For the example illustration, backwards conditioning was undertaken in or-
22
Figure 12: Conditional probability density function for decision information variable cost givenan exposure condition value. The probability density function for the characteristic exposureconditions, original and updated, is shown at 1/1000 scale.
der to obtain a more detailed characterization of the influence of the exposurecondition, 80-year rise in mean annual temperature, on decision information.The results of backwards conditioning may be used to assess the effect of chang-ing the characteristic exposure distribution to favor higher degrees of warming.As shown in Fig. 12, the decision information distributions were similar whenconditioned on the range of exposure values. The irregularities in the surface re-flect the variations in repair timing probabilities shown in Fig. 9. The similarityof the backwards-conditioned decision information distributions suggested that,for this illustration, a change in the exposure distribution would not greatlyaffect the decision information distributions, as was suggested by Fig. 11.
PBDE with convolution allows updating of one pinch-point distributionwithout requiring new analyses in other stages. An updated exposure distri-bution shown in Fig. 5 represented the hypothetical development of betterclimate models or evidence of high emissions. Updated decision informationdistributions were calculated by re-convolving conditional distributions withthe updated exposure distribution, and are shown in Fig. 8. As expected,the updated scenario resulted in negligible changes to the cost and downtimedistributions and decision metrics contained in Table 3.
6. Discussion
6.1. Illustration results and decision support
Results of the format shown in the example illustration could be used tobetter understand the likely deterioration of the structure and to improve theowner’s inspection and repair scheme. The convolution approach predicted a75% probability of corrosion initiation during the lifetime, which suggests thatthe concrete quality and cover depth were insufficient for the exposure. However,the inspection strategy led to repair before significant bar loss or delamination.Assessment of the repair results revealed that the most frequent contributors tothe lifetime cost and downtime were repairs occurring during the first 30 yearsof service. The frequent multiple-repair scenarios were responsible for most ofthe probability of high costs and downtime. An owner wishing to reduce lifetimeimpacts might consider altering the repair strategy to delay repair, or use of acathodic protection system that does not require removal of the concrete cover.Variation in the functional lifetime of the cathodic protection system was notconsidered, and would increase the uncertainty in cost and downtime associatedwith use of cathodic protection for periods greater than 20 years [47].
More generally, the decision information metrics computed by PBDE as-sessment allow improved selection of new designs or maintenance and repairschemes. In the illustration, while the expected value result accurately reflectedthe central tendency of the decision information distributions, it yields only one
23
decision metric. The computed decision information distributions allow use ofnumerous metrics, ranging from means or medians to critical exceedence val-ues. A decision-maker might select an alternate design based on a lower costvalue at 10% exceedence or a lower probability of exceeding some predeterminedcost limit. These probabilistic decision metrics are expected to be more usefulthan a deterministic result, as they support risk-averse or risk-seeking strate-gies [31; 47]. While the computation of decision information distributions isnot exclusive to PBDE, expected values for costs are more frequently used, e.g.,in [13; 27; 28; 30; 47]. Additional decision support can be achieved throughsensitivity assessment, for example by studying the impact of the damage mea-sure thresholds on decision information. Multi-attribute decision optimizationapproaches, such as those discussed in Sec. 2.1.2 could be combined with sen-sitivity assessment to provide further decision support.
6.2. Comparison of convolution and Monte Carlo approaches
The main study indicated that the convolution and Monte Carlo approachesyielded statistically equivalent results at the same number of damage/repairsimulations. Certain techniques, e.g., fitting parametric distributions for re-pair and deterioration conditional distributions, could potentially reduce therequired number of convolution simulations. Combination of Monte Carlo ex-posure/deterioration/repair simulation with analytical computation of decisioninformation distributions may have allowed reduction of the number of MonteCarlo simulations.
While both the convolution and Monte Carlo approaches are modular, theconvolution approach offers some advantages. Assessment of the role of expo-sure on deterioration, repair, and impact was made through easily accessibleintermediate convolution results, but would require additional analysis for theMonte Carlo approach. Similarly, deaggregation and backwards conditioningresults were obtained by manipulating the convolution integral but are difficultto obtain with high robustness using Monte Carlo. Updating of results basedon changes to exposure was simply computed using convolution, but would re-quire bootstrapping or additional simulation for the Monte Carlo approach. Thedeaggregation illustration indicated negligible influence of the exposure condi-tion on decision information, which made updating and re-convolution super-fluous. However, the lack of sensitivity to mean annual temperature increasesuggests the possibility of secondary conditioning on the mean annual temper-ature noise term to determine if that term had a larger effect on the decisioninformation results. Such secondary conditioning can be easily incorporated intothe convolution integral in Eq. 1 through the addition of a secondary exposureconditional distribution.
6.3. Sensitivity of damage and decision information to climate change
While sensitivity of chloride ingress and corrosion to climate change in thisillustration was similar to that seen in other studies, e.g., [3; 6], decision out-comes may differ. These studies found up to a 15% earlier onset of damage or
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reduced service life when climate change from approximately 2000 to 2100 isconsidered. Bastidas-Arteaga et. al [6] finds larger variance in sensitivity, from1.7 to 31% reduction in mean time to failure determined through structural reli-ability analysis, depending on the severity of climate change, the exposure, andthe corrosiveness of the local environment. Results from this study agree thatcorrosion initiation and cracking times may be reduced when climate change isconsidered. However, the extension to repair and impact analysis herein sug-gests that a reduction in apparent service life does not necessarily result in anincrease in impacts due to the mitigating effect of current and presumed futurerepair strategies. Stewart and Peng [5] performed life cycle costing of vari-ous adaptation measures and found that while increasing concrete cover depthslows the onset of deterioration, it is not a rational strategy from a life-cyclecosting perspective. All studies advise that their results do not provide con-clusive prediction of the impact of climate change on corrosion in reinforcedconcrete. More work is needed to assess how accelerated deterioration, repair,and impacts interact before general conclusions can be drawn on appropriateclimate adaptation strategies.
In this example, uncertainty in repair timing and the variances of conditionaldecision information distributions had a more significant influence than climatechange severity on the final impact distributions. For the illustration exam-ple, more research and modeling effort should be directed towards decreasingepistemic uncertainty in conditional decision information models, as the un-certainty in repair timing is thought to reflect natural variability in structuraldeterioration. Inclusion of spatial variation of deterioration parameters, ratherthan consideration of deterioration parameter uncertainty over multiple simu-lations, might allow for a reduction of the number of required simulations. Useof more sophisticated deterioration models would be expected to increase theuncertainty in damage and repair if the sophistication of exposure models wasalso suitably increased, as suggested in [6].
6.4. Modeling extensions for the illustration example
In each analysis stage the models selected aimed for a reasonable degree ofaccuracy and low computational expense; use of different models or assumptionsmay have significantly changed the results. Certain assumptions facilitated ac-curacy in the analysis, including the vertical orientation of the sheltered surface,location of the site in a tropical region, and selection of a single deteriorationmechanism. Other assumptions, such as the use of ambient meteorological dataas exposure conditions, and the use of generic decision information, may havedecreased the potential for accuracy. Important areas for future sensitivity anal-ysis achievable within the convolution approach follow.
The exposure analysis approach ignored many nonlinearities, including pos-sible increases in local weather variability caused by global climate change. How-ever, the deterioration models used were insensitive to fluctuations in temper-ature and relative humidity at short time scales, which decreased the effect ofomitting the complex nonlinearities. The assumption of constant relative hu-midity and chloride exposure with increasing mean annual temperature has been
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widely used, but is controversial, especially considering regional scale climatechanges [49]. Use of a more sophisticated heat, moisture, and ion transportmodel would enable the assessment of the impact of the assumptions of station-ary variability, relative humidity, and chloride exposure on decision information.
Multi-physics models for corrosion in reinforced concrete have been devel-oped, e.g., [35; 50–52], which should be capable of assessing sensitivity to climateparameters. These models also close the deterioration analysis loop by linkingcracking to boundary conditions and transport, which render them more ef-fective for lengthy inspection intervals. On the other hand, empirical models,such as the dose-response models and factor methods [53] used for assessmentof building envelopes, are computationally efficient, which allows for greaterconsideration of uncertainty, e.g., in material properties. Both multi-physicsand empirical approaches can be improved by consideration of spatial variationof damage. Accounting for partial spatial correlation would require the use oftwo- or three-dimensional models, additional damage/repair simulations, boot-strapping, or correlation functions to simulate the states of multiple structuralareas. Techniques for reducing computational expense, such as those discussedin Section 3.6, are likely necessary for implementation of advanced models andpartial spatial correlation in PBDE assessment.
Assumptions made in repair analysis complemented the level of sophistica-tion of deterioration modeling, but could be made more realistic. Repairs wereassumed to occur immediately following inspection. Use of stochastic delay ofa repair following an investigation, or stochastic damage measure thresholds tomimic imperfect inspection techniques, would be straightforward to implementand would increase realism. More significantly, the necessity of maintainingsafety and the prevalence of life-cycle costing suggest considering predictions offuture deterioration and cost when selecting an appropriate repair. Safety riskswere unlikely in this example, but a more sophisticated analysis would need toinclude a structural analysis and reliability model.
Finally, impact analysis could be improved through the use of more sophisti-cated cost, downtime, and environmental impact modeling, e.g., process-basedapproaches [37; 38]. These approaches would mitigate the error associated withindirect applicability of repairs made to different structures, inflation, and lo-cal variations in prices and practices, but take substantial effort to implement.More readily made improvements would include consideration of cost and down-time for the inspections themselves, study of the impact of correlated decisioninformation variables, and use of net-present-value calculations.
6.5. Future work
In addition to the modeling extensions discussed in Section 6.4, researchis required to further develop the PBDE framework in several areas. Notably,work is needed to demonstrate the incorporation of multiple deterioration mech-anisms, and to implement techniques to increase computational efficiency. Theseefforts would prove useful in an attempt to validate the framework on a casestudy structure. Successful validation on a case study structure would demon-strate the viability of the models used in the analysis stages and the framework
26
more generally. Sensitivity assessment or updating using the case study resultscould demonstrate the value of the convolution approach.
The convolution approach’s advantages of deaggregation, backwards con-ditioning and updating would likely hold for other analyses with no couplingbetween analysis stages (i.e., valid conditional independence) and assessmentswhere the assumption of de-coupling is reasonable. For example, assessmentsof bridge scour, building facade damage, or deterioration of steel infrastructureallow the same general assumptions of conditional independence. While theassessment of an infrastructure network where repair decisions are made basedon the status of a number of structures would be computationally challenging,such an assessment would meet the requirements of the convolution approach.
Some assessments may violate the assumption of conditional independencebetween analysis stages. If, for example, a study assumes that the cost of futureenergy depends on the severity of climate change, exposure and impact analysiswould no longer be independent. In such a case, it may be acceptable to assumeconditional independence and make consistent assumptions in computing expo-sure condition and conditional decision-information distributions. Alternately,Monte Carlo simulation may be used to ensure accuracy. If the Monte Carlo ap-proach is used, it is still possible to perform updating or sensitivity assessment,although these tasks become more difficult. Strict requirement of conditional in-dependence hinders the implementation of PBDE in “integrated assessments,”which are becoming more common in climate change impact studies. Whileintegrated assessments allow study of the interaction of policy, climate, and in-frastructure performance, they require feedback loops between analysis stages,thereby reducing the conceptual clarity offered by the PBDE framework.
Suggested applications of the PBDE framework include studies of archetypestructures for use in design code calibration, climate change adaptation pol-icy studies, and sensitivity studies to direct future research. Previous effortshave utilized performance-based approaches to “rationalize” seismic rehabil-itation provisions [20]; similar efforts could be undertaken using the PBDEframework. Planning for climate change adaptation requires a multi-attributedecision framework, and must contend with the high level of uncertainty as-sociated with future emissions, climate response, and climate change impacts.The PBDE framework can accommodate these desirable features, and PBDE’scomputational expense may be acceptable for major regional studies. Finally,the modularity of the PBDE framework supports testbed comparison of mod-els used in the analysis stages, e.g., deterioration analysis, for which there aremany available models and limited data on the decision implications of modelselection. Due to the high computational expense of PBDE, other methods,such as Markov chains or artificial intelligence techniques, may be preferredfor network-level optimization. Similarly, routine decision-making for multi-mechanism deterioration of common structure types may be better supportedby service-life prediction methods.
27
7. Conclusions
A modular framework for performance-based durability engineering (PBDE)assessment has been proposed. The methodology is similar to those of frame-works developed for a variety of engineering risk assessments, and uses modelsdeveloped for service life and durability assessment. By linking uncertainty inanalysis of exposure, deterioration and repair, and impacts, this methodologycan assess the economic, environmental, and social sustainability of structuresin a fully probabilistic manner. Additionally, by comprehensively modeling fromexposure to impacts, the framework is sufficiently general to assess new structuretypes and materials, and is accommodating of shifts in weather patterns causedby climate change. When applicable, the proposed convolution-based method-ology for PBDE offers improvement over traditional Monte Carlo simulation inimplementing deaggregation and backwards conditioning, which allow identifi-cation of important sources of uncertainty. Additionally, PBDE’s modularityfacilitates updating of results to reflect new data or models.
Future work to address the interaction of deterioration mechanisms and thecomputational expense associated with the methodology is needed to supportthe application of PBDE in three targeted domains. PBDE’s consideration ofuncertainty from exposure to impacts may aid design code calibration, espe-cially when considering the potential effects of climate change on structuraldurability. The PBDE framework’s prediction of probabilistic distributions foreconomic, social, and environmental sustainability metrics is thought advan-tageous in guiding policy, e.g., climate change adaptation planning. Finally,flexibility in choice of models allows researchers to assess the effects of modeland parameter selection to direct future work. The proposed PBDE frameworkhas a clear, modular structure, efficiently combines multiple sources of uncer-tainty, and supports analysis of the influence of various uncertainty sources ondecision results.
Acknowledgements
The authors thank Mette Geiker and Alexander Michel for their assistancewith deterioration modeling and developing reasonable deterioration and re-pair assumptions. Jan-Magnus Østvik, Claus Larsen, Arne Gussias, RagnhildRelling, and Knut Grefstad are acknowledged for their help in selecting repairstrategies and developing estimates of repair impacts. Hans Janssen is thankedfor sharing his Latin hypercube optimization routine. Additionally, this workwas made possible by the following funding: the National Science FoundationGraduate Research Fellowship Program and Nordic Research Opportunity, theStanford Gabilan Graduate Fellowship, the John A. Blume Earthquake Engi-neering Center at Stanford University, and the UPS Foundation at StanfordUniversity.
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Appendix A. Deterioration models
The chloride ingress model used is a nonlinear apparent diffusion model sim-ilar to the apparent diffusion model in [24], and to the ion diffusion equation in[6]. The nonlinear influence of chloride binding on diffusion was modeled as in[36], requiring use of numerical methods to solve the partial differential equa-tion [45]. Whereas [36] solves coupled partial differential equations for heat andmass transport, this study neglected ion transport caused by migration and con-vection and assumed instantaneous heat and moisture equilibrium with surfaceconditions. This results in a partial differential equation governing apparentchloride diffusion, Eq. A.1, where CCl is the chloride concentration and DCl
the apparent diffusion coefficient. As reported in [36] the assumption of instan-taneous heat and moisture equilibrium is not generally valid and may lead toerrors in prediction of chloride ingress. However, chloride ingress models thataccept non-stationary boundary conditions are scarce, and fully-coupled heatand mass transport models were prohibitively computationally expensive for adual convolution/Monte Carlo assessment. Thus the dependence of the chloridediffusion coefficient on concrete absolute temperature T , relative humidity h,chloride content, and aging was maintained as shown in Eq. A.2. The referenceapparent diffusion coefficient, D0
Cl, was taken from [24] for ordinary portlandcement concrete with water to cement ratio of 0.5. The temperature dependencetook the form of an Arrhenius rate law, Eq. A.3, where the activation energy,U , was taken from [24]. Moisture-dependency was modeled according to Eq.A.4, also used in [6]. The effect of chloride binding on the diffusion coefficientwas modeled using a Langmuir binding isotherm, Eq. A.5 [36], where CCl isexpressed in moles per cubic meter, assuming 350 kg cement content. Agingwas accounted for using Eq. A.6 from [24]. Parameters for chloride ingress andall other required material parameters are given in Table A.4.
∂CCl∂t
= ∇ (DCl∇CCl) (A.1)
DCl(T, h, CCl) = D0ClfT (T )fh(h)fbind(CCl, θl)fage(t) (A.2)
fT (T ) = exp
(U
R
(1
Tref− 1
T
))(A.3)
fh(h) =
(1 +
(1− h)4
(1− hc)4
)−1(A.4)
fbind(CCl, θl) =1
1 + αbind/ (θl(1 + βbindCCl)2)(A.5)
fage = max((t0/t
)m, 0.1
)(A.6)
Some of these equations require translating concrete relative humidity first tocapillary pressure, pc, according to Eq. A.7, where ρl is the mass density of liquidwater, R the gas constant, and Ml the molar mass of water. Capillary pressurewas converted to moisture content by volume, θl, according to a moisture storage
29
function of the van Genuchten type, Eq. A.8. θcap is the maximum capillarymoisture content, ai, ni and mi are shape parameters and li a weighting factor.The moisture content can be expressed relative to concrete mass, w, throughEq. A.9.
pc =−ρlRTMl
lnh (A.7)
θl = θcap
k∑i=1
li(1 + (aipc)ni)
mi(A.8)
w = θl1/ρl
1/ρconc(A.9)
The partial differential equation (Eq. A.1) was solved in one dimension us-ing a 1 mm uniform mesh from 0-65mm, 5mm mesh from 70-80 mm, and 10mmmesh from 90-250mm depth. The initial condition was no chloride content,the external boundary condition was a Dirichlet boundary of the surface chlo-ride concentration function, and the inner boundary condition was a Neumann“no flux” condition. The transport output time step was 0.04 years. Internaltransport time steps were chosen by the solver.
Corrosion modeling took place at each transport output time step. Rein-forcement corrosion initiation was tested against a stochastic value of the criticalchloride content, CcrCl, which was modeled as lognormally distributed with meanand standard deviation in the range used in [3; 6; 24]. Assumption of a constantcritical chloride content for each simulation is a simplification of a complex phe-nomena, but matched the level of sophistication of the other deterioration mod-els [54]. Reinforcement corrosion current density, icorr [µA/cm2], was calculatedusing a commonly-used empirical model from Liu and Weyers [55] which consid-ers the effect of chloride concentration at rebar depth [kg/m3], temperature [K],concrete resistance, Rc [Ω], and time since corrosion initiation, tc [yr]. It alsoincludes an aleatoric component, leading to a lognormally distributed randomvariable for icorr. The corrosion rate model assumes sufficient oxygen availabilityfor the anodic reaction, which is reasonable given the assumed concrete relativehumidity and moisture content [42; 56; 57]. Concrete resistance was assumedconstant at 25 kΩ, which reflected a moderate degree of capillary saturation [57].Concrete resistivity changes by orders of magnitude depending on the moisturecontent, which would suggest the use of a moisture-dependent resistance value,but Eq. A.10 likely considers moisture content indirectly through the chlorideand temperature dependence, and this effect was omitted.
ln 1.08icorr = 7.89 + 0.7771 ln (1.69CCl)− 3006/T − 0.000116Rc +
2.24t−0.215c +N (0, 0.3312) (A.10)
Reinforcement cross section reduction was calculated from Faraday’s law, Eq.A.11. Here MFe is the molar mass of iron, zFe is the charge number for iron in
30
the corrosion reaction, F is Faraday’s constant, and ρsteel is the density of steel.zFe of 2 was assumed based on the thermodynamic considerations presentedin [56] and reflecting the formation of Fe2O3 at the anodic site. Numericalintegration assumed constant icorr over the time intervals.
∆r =
∫icorrMFe
zFeFρsteeldt (A.11)
Due to the computational expense associated with sophisticated methods ofpredicting delamination or cracking, a simplified approach was used. Coastalexposure can lead to the formation of aqueous rust products that may diffuseaway from the reinforcement without causing a buildup of pressure, furthercomplicating the prediction of cracking. In this analysis it was assumed thatthe rust product formed was Fe2O3, hematite, which is not aqueous, and has avolume ratio to iron of approximately 2.1 [56]. Based on empirical results forordinary Portland cement concretes of similar assumed w/c ratio, bar radiusloss to produce delamination or cover cracking was modeled as lognormallydistributed with mean of 40 µm and standard deviation of 10 µm [58]. Aswith assumptions regarding corrosion initiation and rate, the use of a stochasticdelamination prediction greatly simplifies the complex kinetics and mechanicsof crack formation, but was selected as a reasonable assumption for illustrationof the PBDE methodology.
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Table A.4: Parameters for heat and mass transport, corrosion, and delamination modelsModel Parameter Symbol Value Unit Source
Initial initial relative humidity hinit 0.74 [-] -conditions initial absolute temperature Tinit 295 [K] -
Chloride chloride diffusion coefficient D0Cl 15.8e-12 [m2/s] [24]
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σ∆rcr 10 [µm] [58]
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